TITLE: Why the underlying function of a monomorphism may not be an injection
QUESTION [12 upvotes]: In category theory, a monomorphism (also called a monic morphism or a mono) is defined to be a left-cancellative morphism. It seems that this definition generalizes the definition of injections. However, even in a concrete category, a monomorphism may not be an injection. Why could this happen? I know examples of monomorphisms which are not injections but what is the reason behind the existence of such examples? Isn't monomorphism a generalised concept of injection? Similar questions can be asked about epimorphisms and surjections.

REPLY [5 votes]: I would like to add some remarks (although there is already an accepted answer).
A morphism $f : X \to Y$ is a monomorphism precisely iff $f_\*(T) : \mathrm{Hom}(T,X) \to \mathrm{Hom}(T,Y)$ is injective for all $T \in C$, iff $f_\* : \mathrm{Hom}(-,X) \to \mathrm{Hom}(-,Y)$ is a monomorphism in the category of presheaves $\widehat{C}$. This leads to the following intuition: The notion of monomorphism is only well-behaved if there are enough presheaves.
If $U : C \to D$ is any functor, of course it may fail to preserve or reflect monomorphisms. However, if $U$ has a left adjoint, then it is easy to verify that $U$ preserves monomorphisms. Moreover, when $U$ is faithful, then it is easy to check that $U$ reflects monomorphisms. Thus, if $(C,U)$ is a concrete category over a category $X$ (a category $C$ equipped with a faithful functor $U : C \to X$; see Section 5 in The Joy of Cats), such that $U$ has a left adjoint, then a morphism $f$ in $C$ is a monomorphism iff $U(f)$ is monomorphism in $X$. This applies in particular to the well-behaved case that $C$ is algebraic over $X$.

REPLY [2 votes]: Note that monomorphism is not the only way to generalise an injective map.  There are stronger notions, like a section.  A morphism $f:A\rightarrow B$ is a section if $\exists g:B\rightarrow A$ such that $f\circ g = 1_B$, where $1_B:B\rightarrow B$ is the identity morphism.  Moreover, there also are different kinds of monomorphisms (regular monomorphisms, extremal monomorphisms), most of which are generally stronger notions than a monomorphism, but weaker than a section.  The situation with epimorphisms is dual.